ENGRD 2700: Basic Engineering Probability and Statistics
Spring 2023
Prof. M. Chazal and S. Henderson
Homework 5
Due Wednesday, March 29th at 11:59pm.
ˆ The general guidelines for this homework are the same as for Homework 1.
ˆ The main topics of this homework are Special Continuous Random Variables and Multiple Discrete
Random Variables. A good strategy before starting working on it is:
– To review the lecture notes on Special Continuous Random Variables and Multiple Discrete Random Variables.
– To make sure you have a good understanding of the problems treated in Recitation 7 and 8.
Questions
1. ((9 points) + 2 bonus) You have recently acquired refined glass beads from the Corning Museum
of Glass. For jewelry purpose, it is recommended to drill holes of diameter between 0.99 and 1.01
millimeters (mm), into these 8mm diameter beads. Your power drill produces holes with diameter
being normally distributed with mean µ = 1.00mm and standard deviation σ = 0.08mm.
i) You decide to use your power drill on your beads. What is the probability that the hole drilled in
one randomly selected bead has a diameter of the recommended size, that is, between 0.99 and
1.01mm?
ii) If the recommended range of diameter size was (1 − c, 1 + c), for what value of c would the probability that the hole drilled in a bead has a diameter of recommended size be 95%?
Hints: The symmetry property of the standard normal CDF, Φ(−x) = 1 − Φ(x), may be useful here. The command stats.norm().ppf( · ) will calculate the percentiles associated to the
standard normal distribution.
iii) (Bonus: 2 point) Assume that you can borrow a power drill which produces holes with diameter
being normally distributed with mean µ = 1.00mm but with standard deviation σ < 0.08. For
what values of σ would it be worth borrowing it, if you wanted at least 95% of all drilled beads
to have a diameter of size between 0.99 and 1.01mm?
2. (16 points) An electronic component subject to accelerated life test has a lifetime X with a gamma
probability density function, with mean 24 weeks and standard deviation 12 weeks.
i) What are the values of α and β, shape and scale parameters?
ii) What is the probability that the lifetime is between 12 and 24 weeks?
iii) What is the probability that the lifetime will last at most 24 weeks?
iv) Based on the previous question, is the median of the distribution less than 24? Explain.
v) What is the 99th percentile of the distribution?
vi) The test will be terminated after t weeks. What value of t guarantees only 0.5% of all components
will still be operating at termination?
You may use Python’s function stats.gamma.cdf( · ,α,scale =β) in order to compute values of
the CDF of the gamma distribution with shape α and scale β.
The command stats.gamma.ppf( · ,α,scale =β) will calculate the percentiles associated to the
gamma distribution.
3. (15 points) Let X be the total veterinary expenses Ariana’s Great Pyrenees dog will incur in 2023 as
measured in thousands of dollars. (So, e.g., if X = 1 then the expenses are $1000.) Suppose these
expenses are well approximated by a continuous random variable having PDF f (x) = a(1 + x)−5 , for
x ≥ 0.
i) What is the value of the constant a?
ii) What is the expected value of the veterinary expenses? Hint: This involves an integral. The
change of variable y = 1 + x may make that integral seem a bit easier.
iii) We know that E(X 2 ) = 31 . What is the standard deviation of the veterinary expenses?
iv) Find the CDF of X, FX (x).
v) Ariana’s veterinary insurance policy has a $500 deductible (so Ariana has to pay the first $500
of the expenses). The plan then pays 90% of additional expenses beyond the deductible. The
policy also allows for a maximum out-of-pocket expense of $1000, i.e., Ariana will not pay more
than $1000 for veterinary expenses because the insurance company will cover her bills at 100%
once her share of the expenses reaches this value. Let Y denote the share of Ariana’s veterinary
expenses that she will pay, also measured in units of $1000. Express Y in terms of X.
vi) Find the CDF of Y , FY (y).
4. (10 points) A box contains the beads you got from the Corning Museum of Glass in Question 1. They
are of three different colors. Of all beads, one hundred are Azure, two hundred are Byzantine, and
three hundred are Coquelicot. Two beads are drawn at random and with replacement. Let X (resp.,
Y ) be the number of Azure (resp., Byzantine) beads among the two selected at random.
i) Complete the table of the joint PMF pX,Y (x, y) below.
pX,Y (x, y)
0
Y
1
2
0
12
· 4
502
1
· 4
502
1
· 4
50
X
1
12
· 4
502
1
· 4
50
0
2
12
· 4
50
0
0
ii) Find the marginal PMFs.
iii) Are the random variables X and Y independent? Justify.
iv) Compute Cov(X, Y ).
5. (12 points) The number of Cornell logo backpacks sold by the Cornell Store in the month of April is
a random variable X with PMF given by
x
pX (x)
0
0.15
1
0.25
2
0.3
3
0.20
4
0.1
Sixty percent of backpack purchasers also purchase a laptop sleeve. Let Y be the random number of
sleeves bought by backpack purchasers in April.
i) What is P (X = 4, Y = 2)? (Think conditional probability)
ii) Compute P (X = Y ).
iii) What is the joint PMF of (X, Y )?
iv) What is the marginal PMF of Y ?